Editing Radiometry (section)

== Integral and spectral radiometric quantities ==
[[Integral]] quantities (like [[radiant flux]]) describe the total effect of radiation of all [[wavelength]]s or [[frequency|frequencies]], while [[electromagnetic spectrum|spectral]] quantities (like [[spectral power]]) describe the effect of radiation of a single wavelength {{mvar|λ}} or frequency {{mvar|ν}}. To each '''integral quantity''' there are corresponding '''spectral quantities''', defined as the quotient of the integrated quantity by the range of frequency or wavelength considered.<ref name="ISO 2013 i869">{{cite web | title=ISO 80000-7:2019 - Quantities and units, Part 7: Light and radiation | website=ISO | date=2013-08-20 | url=https://www.iso.org/standard/64977.html | access-date=2023-12-09}}</ref> For example, the radiant flux Φ<sub>e</sub> corresponds to the spectral power Φ<sub>e,{{mvar|λ}}</sub> and Φ<sub>e,{{mvar|ν}}</sub>.

Getting an integral quantity's spectral counterpart requires a [[Limit (mathematics)|limit transition]]. This comes from the idea that the precisely requested wavelength [[photon]] existence probability is zero. Let us show the relation between them using the radiant flux as an example:

Integral flux, whose unit is [[watt|W]]:
<math display=block>\Phi_\mathrm{e}.</math>
Spectral flux by wavelength, whose unit is {{nobreak|W/[[metre|m]]}}:
<math display=block>\Phi_{\mathrm{e},\lambda} = {d\Phi_\mathrm{e} \over d\lambda},</math>
where <math>d\Phi_\mathrm{e}</math> is the radiant flux of the radiation in a small wavelength interval <math>[\lambda - {d\lambda \over 2}, \lambda + {d\lambda \over 2}]</math>.
The area under a plot with wavelength horizontal axis equals to the total radiant flux.

Spectral flux by frequency, whose unit is {{nobreak|W/[[hertz|Hz]]}}:
<math display=block>\Phi_{\mathrm{e},\nu} = {d\Phi_\mathrm{e} \over d\nu},</math>
where <math>d\Phi_\mathrm{e}</math> is the radiant flux of the radiation in a small frequency interval <math>[\nu - {d\nu \over 2}, \nu + {d\nu \over 2}]</math>.
The area under a plot with frequency horizontal axis equals to the total radiant flux.

The spectral quantities by wavelength {{mvar|λ}} and frequency {{mvar|ν}} are related to each other, since the product of the two variables is the [[speed of light]] (<math>\lambda \cdot \nu = c</math>):
:<math>\Phi_{\mathrm{e},\lambda} = {c \over \lambda^2} \Phi_{\mathrm{e},\nu},</math> or <math>\Phi_{\mathrm{e},\nu} = {c \over \nu^2} \Phi_{\mathrm{e},\lambda},</math> or <math>\lambda \Phi_{\mathrm{e},\lambda} = \nu \Phi_{\mathrm{e},\nu}.</math>

The integral quantity can be obtained by the spectral quantity's integration:

<math display=block>\Phi_\mathrm{e} =
\int_0^\infty \Phi_{\mathrm{e},\lambda}\, d\lambda =
\int_0^\infty \Phi_{\mathrm{e},\nu}\, d\nu =
\int_0^\infty \lambda \Phi_{\mathrm{e},\lambda}\, d \ln \lambda =
\int_0^\infty \nu \Phi_{\mathrm{e},\nu}\, d \ln \nu.</math>